What information is conveyed by the numerical values we get by putting the coordinates of a point in a locus' equation?

Question

If we put the Cartesian coordinates of a point in a 2 dimensional locus' equation then we get zero as the value if the point lies on the locus. On putting coordinates of all other points in the equation which do not lie on the locus, we get a numerical value. Does this numerical value convey any information about the position of the point with respect to the locus?

Example: Suppose you have the equation of a straight line as 2x+y=0. If you put the coordinates of the points lying on the line, you get R.H.S. as 0,i.e., the condition is satisfied. However if we put in the coordinates of points(x,y) which lie lower than the line in the L.H.S. (2x+y) , we get R.H.S.<0 and on putting the values of of points that lie above the line, we get R.H.S.>0. Can we derive any relationship between the position of points and the value if R.H.S. we get by putting their coordinates in the equation's L.H.S. ?

Answer 1

If the null set of a function $f:\mathbb R^n\to\mathbb R$ divides the space into two regions, it’s often possible to use the sign of $f(\mathbf x)$ to determine on which “side” of the dividing surface the point $\mathbf x$ lies: all of the points for which $f(\mathbf x)$ has the same sign lie in the same region. As with orientation in general, the choice of which region is left/right or inside/outside is arbitrary.

A basic example is a line $ax+by+c=0$ in $\mathbb R^2$. For an arbitrary point not on this line, if $ax+by+c\gt0$, then it lies on the side of the line toward which the line’s normal $(a,b)$ points; if negative, it lies on the side away from the normal. Similarly, for the circle defined by $f:(x,y)\mapsto x^2+y^2-r^2$, points for which $f(x,y)\gt0$ are in one region (the outside) and points for which $f(x,y)\lt0$ are in the other. This in fact works for a general quadric in $\mathbb R^n$ as well, although defining “inside” and “outside” for these more complex surfaces can take a bit of thought.

Answer 2

Suppose we have the function $f(x,y,z) = x^2 + y^2 - z.$ The locus of all points $(x,y,z)$ where $0=f(x,y,z)$ is a paraboloid of revolution. For a fixed $z=r^2$ the section of the surface, that is, all points $(x,y)$ where $0=f(x,y,r^2)$, is a circle of radius $r$. This example generalizes to other cases where $f(x,y,z)=g(x,y)-z$ and we want the locus of all points $(x,y)$ where $z=g(x,y)$. Note that, if $h(x)$ is an invertible function, then $f(x,y,z) = h(g(x,y))-z$ will describe the same locus. Therefore, there is not much information in $g(x,y)$ in general, but its sign is significant.